Correcting for the alias effect when measuring the power spectrum using FFT

TL;DR

The paper addresses biases in power-spectrum estimation arising from mass assignment and sampling when using FFT. It derives an exact relation between the FFT-estimated spectrum $\big\langle |\delta^{f}(\mathbf k)|^2\big\rangle$ and the true power spectrum $P(k)$, including alias sums and shot-noise terms, and introduces an iterative de-aliasing method to reconstruct $P(k)$ from the alias-laden estimate. For NGP, CIC, and TSC mass assignments, it provides analytic shot-noise corrections and demonstrates, via N-body simulations, that the recovered $P(k)$ agrees across grid resolutions and mass-assignments, extending also to samples with selection effects. This framework enables robust, grid-independent recovery of $P(k)$ on scales up to the Nyquist wavenumber, with practical implications for large-scale structure analyses and survey design.

Abstract

Because of mass assignment onto grid points in the measurement of the power spectrum using the Fast Fourier Transform (FFT), the raw power spectrum $\la |δ^f(k)|^2\ra$ estimated with FFT is not the same as the true power spectrum $P(k)$. In this paper, we derive the formula which relates $\la |δ^f(k)|^2\ra$ to $P(k)$. For a sample of $N$ discrete objects, the formula reads: $\la |δ^f(k)|^2\ra=\sum_{\vec n} [|W(\kalias)|^2P(\kalias)+1/N|W(\kalias)|^2]$, where $W(\vec k)$ is the Fourier transform of the mass assignment function $W(\vec r)$, $k_N$ is the Nyquist wavenumber, and $\vec n$ is an integer vector. The formula is different from that in some of previous works where the summation over $\vec n$ is neglected. For the NGP, CIC and TSC assignment functions, we show that the shot noise term $\sum_{\vec n} 1/N|W(\kalias)|^2]$ can be expressed by simple analytical functions. To reconstruct $P(k)$ from the alias sum $\sum_{\vec n}|W(\kalias)|^2 P(\kalias)$, we propose an iterative method. We test the method by applying it to an N-body simulation sample, and show that the method can successfully recover $P(k)$. The discussion is further generalized to samples with observational selection effects.

Figures (2)

• Figure 1: The shot noise $D^2(\mathbf k)$ estimated from 10 samples of Poisson distributed random points. Each symbol represents the result for each mass assignment, as indicated in the figure. The upper panel shows the result with the function $\langle D^2(\mathbf k)N/C_3(\mathbf k)\rangle_d$, i.e. the $D^2(\mathbf k)$ scaled to $1/N C_1(\mathbf k)$. The estimated result agrees quite well with the analytical prediction $\langle D^2(\mathbf k)N/C_3(\mathbf k)\rangle_d=1$. The lower panel compares the estimated $\langle D^2(\mathbf k)N\rangle_d$ with our analytical predictions for the NGP (solid line), CIC (dotted line) and TSC (dashed line) assignment functions. For CIC and TSC, we have used the approximate formulae of Eq. (21).
• Figure 2: The six power spectra which we measured for an N-body simulation sample using three mass assignments (NGP, CIC and TSC) and two grids ($64^3$ and $256^3$ grid points). $k_N^{64}$ is the Nyquist wavenumber of $64^3$ grid points. The upper panel shows the raw power spectra $\langle|\delta^f(k)|^2\rangle$ estimated directly from the FFT (Eq. 8). The lower panel shows the true power spectra $P(k)$ reconstructed from the $\langle|\delta^f(k)|^2\rangle$ following the procedure described in the text. The six reconstructed $P(k)$ agree so well that their curves overlay each other.